How to quickly change hand-drawn shapes to symmetrical polished shapes?
From how it looks, the most reliable method to try out is using Hough transform.
The Hough transform can be used to detect e.g. lines and circles in images (depending on which variant you are using; in this case it would amount to a combination of variants for both lines and circles obviously).
So, given some input image, the Hough transform tells you what the line/circle parameters are that have created a line in the input image. For example, given a line, it would tell you the intersection with the $y$-axis and the $slope$ of the line detected in the image.
Then, you could use these parameter information to reconstruct lines and circles detected in the image.
The last remaining problem to be solved then is to check where a detected line starts and stops (since this is not obvious from line parameters like $m$ (=$slope$) and $b$ (=intersection with $y$-axis) in the equation $y = mx+b$ describing some line).
But for that, you could &quot;walk&quot; along a line in the image space and check where the line is present or not. Then, you can draw line segments in the reconstruction image when the corresponding elements are also present in the original image.
The problem with (C)NNs would be that they are sensitive to rotation and scale etc. You could of course take a tremendously large number of filters to account for shapes of different rotation and scale, but that would increase the demand for labeled training data again (which could of course be automatically be generated a priori in this simple case).
Anyway, I'd suggest checking out Hough transform. To get some feeling for it, there are lots of libraries available implementing it for Python or MatLab, for example.
For further information, check out Wikipedia or YouTube.